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. 2007 Apr 10;104(15):6266-71.
doi: 10.1073/pnas.0607280104. Epub 2007 Apr 3.

Complete genetic linkage can subvert natural selection

Philip J Gerrish  1 , Alexandre Colato, Alan S Perelson, Paul D Sniegowski
Affiliations

Complete genetic linkage can subvert natural selection

Philip J Gerrish et al. Proc Natl Acad Sci U S A. .

Abstract

The intricate adjustment of organisms to their environment demonstrates the effectiveness of natural selection. But Darwin himself recognized that certain biological features could limit this effectiveness, features that generally reduce the efficiency of natural selection or yield suboptimal adaptation. Genetic linkage is known to be one such feature, and here we show theoretically that it can introduce a more sinister flaw: when there is complete linkage between loci affecting fitness and loci affecting mutation rate, positive natural selection and recurrent mutation can drive mutation rates in an adapting population to intolerable levels. We discuss potential implications of this finding for the early establishment of recombination, the evolutionary fate of asexual populations, and immunological clearance of clonal pathogens.

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Conflict of interest statement

The authors declare no conflict of interest.

Figures

Fig. 1.
Fig. 1.
Representative plots of observed fitness (blue lines) and mutation rate (red lines) dynamics. (A) A numerical solution of Model 1 (infinite population) with parameters μ = 0.003, fB = 10−6, fD = 0.1, fM = 0.001, fA = 0.0001, and mB = mD = mM = mA = 0.03, and initial condition u(x, y, 0) ≈ δ(x − x*, y − y*), where x* and y* were close to the minimum values allowed for x and y. (B) An individual-based simulation run (finite population) with population size N = 10,000 and parameters μ = 0.1, fB = 3 × 10−4, fD = 0.5, fM = 0.001, fA = 0.0001, and mB = mD = mM = mA = 0.03. Initially, all individuals in the population had fitness zero and a relative mutation rate of one. The thin green line plots predicted fitness based on cumulative fitness variance and mutation rate: x̄(t) = ∫0t σx2dτ − fDmD∫0t μ̄ dτ. The thin pink line plots predicted mutation rate: μ̄(t) = ∫0t cov(μ, x)dτ + fMmM∫0tμ2dτ. Based on the number of fluctuations in cov(μ, x), we estimate that extinction was caused in this population by 27 associations between mutators and beneficial mutations.
Fig. 2.
Fig. 2.
Sensitivity of extinction times (all vertical axes) to various parameters. Red corresponds to Wright–Fisher simulations (SI Text) of populations of size N = 1,000. Green corresponds to regular simulations with population size N = 10,000. Default parameter values were μ = 0.1, fB = 3 × 10−4, fD = 0.1, fM = 0.001, fA = 0.0001, and mB = mD = mM = mA = 0.03, and initial conditions were as in Fig. 1B. Extinction times were plotted against the fraction of mutations that are beneficial (A) and deleterious (B). Open circles indicate populations that only declined in fitness. (C) The ratio of mutator to antimutator fractions. The blue line indicates the theoretical upper bound for infinite populations. (D) Mean effect of beneficial mutations. (E) Mean effect of deleterious mutations. (F) Ratio of mean mutator effect to mean antimutator effect. The blue line indicates the theoretical upper bound for infinite populations. (G) Initial population size. The green line indicates the theoretical maximum value that extinction times should converge to as population size increases further. (H) The “hitchhiking index” (h = cov(μ, x)). The green line plots tE as defined in the text (infinite-population limit).
Fig. 3.
Fig. 3.
Comparing analytical theory with individual-based simulations. Data were taken from a simulation run with a population size of 10,000. (A) Predicted values for ∂μ̄/∂t were obtained by computing cov(μ, x) + fMmMμ2, and observed values for ∂μ̄/∂t were obtained by computing the slope of mean mutation rate. The parameters fM and mM were the same as in the simulations. (B) Predicted values for ∂x̄/∂t were obtained by computing σx2 − fDmDμ̄, and observed values for ∂x̄/∂t were obtained by computing the slope of mean log fitness. The parameters fD and mD were the same as in the simulations.
Fig. 4.
Fig. 4.
The mechanics of extinction. (A) Mutation rate partitioned into its two theoretical components: the hitchhiking component (HC, blue line) and the mutation component (MC, green line). The predicted mutation rate (thin black line) is the sum of these two components, μ̄(t) = ∫0t cov(μ, x)dτ + fMmM∫0tμ2dτ. The red line plots observed mutation rate. Eventually, mutation rate becomes very costly causing HC to decline sharply. After this, MC dominates and more than compensates for the sharp decline in HC. The balance is tipped by hitchhiking (fitness peaks early) but the subsequent divergence of cov(μ, x) and fMmMμ2 is what deals the final and decisive blow. (B) Results of single-lineage simulations. Lineages started with one founding individual with an absolute fitness of 1.7 and a deleterious mutation rate of 0.1 times the relative mutation rate (abscissa). The cutoff time was 1,000 generations. Above the threshold mutation rate, lineages are doomed to extinction. Surprisingly, these condemned lineages nevertheless thrive for a considerable time (green dots) and contribute a large number of cumulative members (red squares) before going extinct. This finding illustrates the delayed fitness consequence of an intolerable mutation rate, and explains how natural selection can cause a population to evolve an intolerable mutation rate and subsequently go extinct.
Fig. 5.
Fig. 5.
Error catastrophe driven by immune response. This is a plot of the numerical solution of Model 2 in Biomedical Implications. Parameter values were taken from previous work on a similar model (47): a = 1.1, d = 0.02, β = 5, and we further assumed μ = 0.1 and σ2 = 0.1 (dynamics were fairly robust to these values).
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